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Geometry of the Projectivization of Ideals and Applications to Problems of Birationality Remi Bignalet-Cazalet

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Geometry of the Projectivization of Ideals and Applications to Problems of Birationality Remi Bignalet-Cazalet Geometry of the projectivization of ideals and applications to problems of birationality Remi Bignalet-Cazalet To cite this version: Remi Bignalet-Cazalet. Geometry of the projectivization of ideals and applications to problems of birationality. Algebraic Geometry [math.AG]. Université Bourgogne Franche-Comté, 2018. English. NNT : 2018UBFCK038. tel-02004276 HAL Id: tel-02004276 https://tel.archives-ouvertes.fr/tel-02004276 Submitted on 1 Feb 2019 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. These` de doctorat de l'Universite´ de Bourgogne Franche-Comte´ pr´epar´ee`al'Institut de Math´ematiquesde Bourgogne Ecole´ doctorale Carnot Pasteur (ED553) Doctorat de Math´ematiques par R´emiBignalet-Cazalet Geom´ etrie´ de la projectivisation des ideaux´ et applications aux problemes` de birationalite´ Directeurs de th`ese: Adrien Dubouloz et Daniele Faenzi Th`esesoutenue le 24 octobre 2018 `aDijon Composition du jury Laurent Buse´ INRIA - Sophia Antipolis Examinateur Julie Deserti´ IMJ-PRG - Universit´eParis Diderot Examinatrice Adrien Dubouloz IMB - Universit´ede Bourgogne Directeur de th`ese Daniele Faenzi IMB - Universit´ede Bourgogne Directeur de th`ese Laurent Manivel IMT - Universit´ePaul Sabatier Rapporteur Lucy Moser-Jauslin IMB - Universit´ede Bourgogne Pr´esidente du jury Fransesco Russo DMI - Universit`adegli Studi di Catania Rapporteur Ronan Terpereau IMB - Universit´ede Bourgogne Examinateur Institut de Math´ematiquesde ´ Bourgogne Ecole Doctorale Carnot-Pasteur UMR 5584 CNRS Universit´ede Bourgogne Universit´ede Bourgogne Franche-Comt´e Franche-Comt´e UFR Sciences et Techniques 9 Avenue Alain Savary 9 Avenue Alain Savary BP 47870 21078 Dijon Cedex BP 47870 21078 Dijon Cedex 1 Al andar se hace camino y al volver la vista atr´as se ve la senda que nunca se ha de volver a pisar. Antonio Machado, Caminante no hay camino 2 3 En marchant se fait le chemin et c'est en se retournant que l'on peut contempler le sentier que l'on n'aura jamais plus l'occasion d'emprunter. Traduction personnelle 4 5 Remerciements Au cours de ma th`ese,j'ai pu b´en´eficierdu concours, du soutien ou de l'aide de personnes que je tiens `aremercier. J'esp`ereque ces quelques lignes traduiront ma profonde gratitude envers eux. Mes premiers remerciements vont `ames deux directeurs de th`ese, Adrien Dubouloz et Daniele Faenzi. Que de patience, que d'indulgence de leur part durant ces trois ann´ees! Je tiens `ales remercier particuli`erement pour leur disponibilit´e et leur exigence lors de toutes les phases r´edactionnellesde mon doctorat. J'ai pu comprendre, gr^ace`aeux, les passages oblig´esd'´elaboration, de maturation et de diffusion d'une id´eemath´ematique.J'ai beaucoup appr´eci´eaussi les discussions passionn´eesd'Adrien sur le fonctionnement du monde math´ematiquequ'il m'a aid´e `amieux envisager et la clairvoyance et les fulgurances de Daniele qui ont ´et´edes sources constantes d'inspiration. Un grand merci aussi `a • Laurent Manivel et Francesco Russo pour avoir rapport´ecette th`ese.Leurs remarques ont permis de grandement am´eliorerle manuscrit initial et m'ont fait prendre conscience de questions que je n'avais pas encore clairement identifi´ees,notamment dans les chapitres 4 et 6, • Julie D´esertiqui m'a aiguill´esur cette th`eseapr`esavoir encadr´emon m´emoire de master. Les th´ematiquesque j'ai abord´eesdurant ces trois ann´eesont ´et´e tr`esinfluenc´eespar ce travail initial et je la remercie d'autant plus d'avoir accept´ed'^etredans mon jury de th`ese. • Laurent Bus´e,Lucy Moser-Jauslin et Ronan Terpereau pour avoir accept´e de faire partie de mon jury de th`eseet particuli`erement Ronan dont j'ai ´enorm´ement pu appr´ecierle dynamisme et l'engagement dans les activit´es math´ematiqueset extra-math´ematiquesdijonnaises. De mani`ereg´en´erale,merci `ala dynamique ´equipe de G´eom´etriealg´ebrique de Dijon, dont Fr´ed´ericD´egliseet Jan Nagel, pour m'avoir fait d´ecouvriret associ´e `atant d'activit´esmath´ematiques,`aJos´e-Luispour sa bienveillance, `aEmmanuel qui m'a mis le pied `al'´etrierde l'enseignement `al'universit´eainsi qu’`aPeggy et Fran¸coispour leur aide pr´ecieusedans certaines de mes d´emarches administratives. Je tiens `aremercier Charlie dont la fiabilit´er´esiste `atoute ´epreuve et ceux qui m'ont rendu la vie de doctorant `aDijon plus agr´eable,les membres de l'IMB et toute son ´equipe administrative, les doctorants ou ex-doctorants de l'IMB, en particulier mes deux anciens co-brasseurs Antoine et J´er´emy avec qui j'ai pass´eune premi`ere ann´eede th`eseparticuli`erement productive... Je tiens `aciter Matilde, Vladimiro, Andr`es,Anne, Mathias, Fabio dont j'ai ´enorm´ement appr´eci´ela compagnie entre autre dans des conf´erencesou ´ecolesde maths. Un merci plus sp´ecial `aMatilde pour m'avoir accueilli dans cette belle ville de San Marin durant quelques jours si particuliers d'ao^ut.Plus g´en´eralement, je remercie tous mes amis, palois, palois d´esormaisparisiens et autres provinciaux parisiens, dijonnais ou anciens dijonnais ainsi que toute ma famille qui ont ´et´ed'un soutien sans faille tout au long de mon doctorat. Dijon, le 10 octobre 2018. 6 R´esum´egrand public L'explosion r´ecente des capacit´esde calculs num´eriquesqui s'est produite au cours des trente derni`eres ann´eesa renvers´eles perspectives de recherches et d´eveloppements. La simulation num´eriquepermet d´esormaisune estima- tion plus fiable et beaucoup plus rapide de r´esultatsautrefois inatteignables. Dans cette th`ese,nous appliquons ce principe tr`esg´en´eral`aun domaine des math´ematiques,la g´eom´etriealg´ebrique,qui concerne l'´etudedes ´equations polynomiales. Cela nous permet de pr´evoir l'existence de configurations g´eom´etriquesinattendues apportant ainsi un point de vue original sur des su- jets classiques de g´eom´etrie.En plus de ces simulations, nous nous attachons `ad´emontrer math´ematiquement ces ph´enom`enesannonc´esnum´eriquement. Nos r´esultatsconcernent aussi le d´eveloppement de m´ethodes num´eriques visant `aam´eliorerdavantage les capacit´eset la rapidit´edes simulations dans ce domaine. Large audience abstract The recent growth in capacity of numerical calculus over the last thirty years has brought a major change in perspective regarding matters of re- search and development. Numerical simulation allows a more reliable and faster estimation of results unattainable beforehand. In this thesis, we apply this very general principle to a domain of mathematics, algebraic geometry which concerns the study of polynomial equations. This allows us to predict unexpected geometrical configurations bringing an original point of view to classical geometrical subjects. In addition of those simulations, we focus on proving mathematically those predicted phenomenons. Our results also con- cern the development of numerical methods aiming to further improve the capacity and the rapidity of simulations in this area. 7 R´esum´e Dans cette th`ese,nous interpr´etonsg´eom´etriquement la torsion de l'alg`ebre sym´etriqued'un faisceau d'id´eaux IZ d'un sch´ema Z d´efinipar n+1 ´equations dans une vari´et´en-dimensionnelle. Ceci revient `a´etudierla g´eom´etriede la projectivisation de IZ . Les applications de ce point de vue concernent en particulier le domaine des transformations birationnelles de l'espace projectif de dimension 3 au sujet duquel nous construisons des transformations bi- rationnelles explicites qui ont le m^emedegr´ealg´ebrique que leur inverse, le domaine des courbes libres et presque-libres au sujet duquel nous g´en´eralisons une caract´erisationdes courbes libres en ´etendant les notions de nombre de Milnor et de nombre de Tjurina. Nous abordons aussi le sujet des hypersur- faces homaloides, notre motivation initiale, au sujet duquel nous exhibons en particulier une courbe homaloide de degr´e5 en caract´eristique3. La derni`ere application concerne le calcul de l'inverse d'une transformation birationnelle. Mots cl´es: G´eom´etriealg´ebrique,Alg`ebrecommutative, Th´eoriedes singularit´es, Transformations birationelles, Hypersurfaces homalo¨ıdes,courbes libres et presque libres, alg`ebrede Rees et alg`ebresymm´etrique,Syzygies, R´esolutions Title of the thesis : Geometry of the projectivization of ideals and applications to problems of birationality Abstract In this thesis, we interpret geometrically the torsion of the symmetric algebra of the ideal sheaf IZ of a scheme Z defined by n + 1 equations in an n-dimensional variety. This is equivalent to study the geometry of the projectivization of IZ . The applications of this point of view concern, in particular, the topic of birational maps of the projective space of dimension 3 for which we construct explicit birational maps that have the same algebraic degree as their inverse, free and nearly-free curves for which we generalise a characterization of free curves by extending the notion of Milnor and Tjurina numbers. We tackle also the topic of homaloidal hypersurfaces, our original motivation, for which we produce in particular a homaloidal curve of degree 5 in characteristic 3. The last application concerns the computation of the inverse of a birational map. Keywords: Algebraic Geometry, Commutative algebra, Singularity theory, Bira- tional maps, Homaloidal hypersurfaces, free and nearly free curves, Symmetric and Rees algebra, Syzygies, Resolutions 8 Contents Introduction 11 Contents of the manuscript .......................... 17 1 Degrees of rational maps 23 1.1 Proj of a sheaf .............................
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